How do you solve #x/2 = sqrt( x - 1)#?

1 Answer
Aug 19, 2015

Answer:

#x = 2#

Explanation:

First, start by taking a look at the equation

#sqrt(x-1) = x/2#

Right from the start, you need #x# to be positive. Not only that, but you need the expression that's under the square root to be positive as well, since you cannot take the square root of a negative number and get a real number as a solution.

So, you know that you need

#x - 1 >=0 implies x>=1#

Next, square both sides of the equation to get rid of the square root

#(sqrt(x-1))^2 = (x/2)^2#

#x-1 = x^2/4#

This is equivalent to

#x^2 -4x + 4 = 0#

Use the quadratic formula to find the solutions to this quadratic equation

#x_(1,2) = (-(-4) +- sqrt((-4)^2 - 4 * 1 * (-4)))/(2 * 1)#

#x_(1,2) = (4 +- sqrt(0))/2#

This means that the quadratic has one distinct solution

#x = 4/2 = color(green)(2)#

Since #x =2# satisfies the condtion #x>=1#, this will also be the solution to your original equation.

You can do a quick check to make sure that the calculations are correct

#sqrt(2 - 1) = 2/2#

#sqrt(1) = 1#

#1 = 1" "color(green)(sqrt())#